unidirectionally multispecies transitions

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PHYSICAL REVIEW E 71, 046122 2005

Absorbing phase transitions of unidirectionally coupled nonequilibrium systems

Sungchul Kwon* and Gunter M. Schütz

Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany

Received 11 October 2004; published 18 April 2005

We investigate the change of critical behavior of two-level hierarchy systems in which the second level B

is unidirectionally coupled to the first A by the coupling dynamics A→A+nB with n=1 or 2. The first level

belongs to the directed percolation or the parity-conserving PC universality class, the second to PC. If both

levels are critical, the active region of the second level becomes heterogeneous. In the so-called coupled region

the first level feeds particles to the second, while in the uncoupled region the second level evolves autonomously.

Measuring dynamic critical exponents in both regions, we show to what extent the critical behavior of

the second level depends on the universality class of the first. These results suggest a simple criterion for the

emergence of unusual critical behavior of unidirectionally coupled nonequilibrium systems.

DOI: 10.1103/PhysRevE.71.046122

PACS numbers: 64.60.i, 05.40.a, 82.20.w, 05.70.Ln

I. INTRODUCTION

Recently there has been an enormous effort to understand

nonequilibrium continuous absorbing phase transitions

APT’s from an active into absorbing states in which no

further change occurs 1,2. The concept of an APT can be

applied to various phenomena in natural science as well as

social and economical science. From a conceptual point of

view APT’s can be regarded as one of the simplest and most

natural extensions of the well-established equilibrium phase

transitions to nonequilibrium systems. As in equilibrium, the

dimensionality of a system, symmetries between absorbing

states and conservation laws may determine critical behavior

of APT’s. However, as a unifying theoretical framework is

not available, we are still far from a systematic classification

of APT’s. Exact non-mean-field critical exponents and scaling

functions have been derived only for systems with a

singular APT where the density in the active phase diverges

see 3,4 for recent progress.

Few universality classes of continuous APT’s have been

identified so far. The directed percolation DP and parityconserving

PC universality class are well established

among others 2,5. The DP class includes systems which

have no symmetry between absorbing states and no conservation

laws of order parameters 1,2, while the PC class

describes models with a modulo 2 conservation of the total

number of particles via reactions A→1+2kA and 2A→0

defining branching annihilating random walks BAW’s with

an even number of offspring 5. The so-called directed Ising

DI class includes models with two symmetric absorbing

states 6. In1+1 dimension the DI and PC classes are

characterized by the same exponents 2,6.

Unusual critical behavior has been observed in various

coupled systems of known universality classes 2,7–17. In

coupled systems one linear case or more nonlinear case

particles of one species give birth to an offspring of another

species. However, in some cases the universality class does

*Present address: Department of Physics, Kyung Hee University,

Seoul 130-701, Korea.

not change. For example, quadratic and bidirectional coupling

of DP processes exhibit the usual DP-type critical behavior

7. Also linearly and bidirectionally coupled DP systems

exhibit either DP or trivial critical behavior 9

depending on the existence of the spontaneous annihilation

of a single particle, A→ 10. The same type of coupling

of PC processes makes systems always active so only the

critical behavior near the zero branching rate is observed 8.

On the other hand, cyclic coupling of DP processes and annihilating

random walks may result in nontrivial critical behavior

depending on a coupling constant 11. The cyclically

coupled model corresponds to the multispecies mapping of

pair-contact process with diffusion 19, the universality

class of which is much disputed.

Linear and unidirectional couplings have been first considered

in recent studies on layer-by-layer interface growth

models for adsorption and desorption of monomers 12. It

turns out that the unidirectional coupling dynamics from

lower to higher layers are key features characterizing the

continuously varying critical behavior of the monomer

growth models 12,14. Some polynuclear growth models

and models for fungal growth also exhibit critical behavior

of the unidirectionally coupled DP processes 15. Using

bosonic field theoretic renormalization group techniques,

Täuber et al. extensively studied the hierarchy of unidirectionally

coupled DP subsystems, which is a stochastic multispecies

particle system 13,14. Coupled PC processes have

been studied in the context of interface growth models which

conserve the parity of the particle number in each layer. Such

dimer models undergo two different types of phase transitions,

roughening and faceting transitions 17,18. At the

roughening transition, the critical behavior of the bottom

layer belongs to the PC class, whereas higher layers exhibit

continuously varying critical behavior.

In general, a linearly unidirectionally coupled system can

be defined as a hierarchy of a number of levels belonging to

known universality classes. Each level is coupled to the next

level by branching dynamics of the form

A i → A i + nA i+1 ,

1

with a positive integer n. While direct couplings with any

higher levels are possible, such as A 1 →A 1 +A i , it was shown

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that they are irrelevant in determining the critical behavior

13,14,17.

So far we have discussed coupled DP or PC processes. In

this paper, we continue to investigate the properties of the

linearly unidirectionally coupled systems in which each level

can belong to a different universality class. For selfcontainedness

we briefly review the usual definition of critical

exponents and discuss the heterogeneity of active regions

and the source average, the significance of which were detailed

in recent work 16 Sec. II. We then discuss the twolevel

hierarchies. We call the first level source and the second

level slave and classify the couplings between two

classes as class of source–class of slave couplings according

to the class of each level. In our previous study 16

we considered the cases where the source level belongs to

either the DP or PC universality class and the slave level is in

the DP class. Here we discuss the DP-PC Sec. III and

PC-PC Sec. IV couplings, which are not investigated in

16. In Sec. V we present simulation results of other couplings.

In the concluding section VI we summarize our main

findings from which we postulate a simple general criterion

for the critical behavior of the slave level for general couplings.

II. CRITICAL BEHAVIOR AND AVERAGING IN

UNIDIRECTIONALLY COUPLED SYSTEMS

A. Critical exponents in coupled APT’s

As in the usual APT’s, several critical exponents characterize

the off-critical and critical behavior of the coupled

systems. Near criticality the exponents , , , and

characterize the scaling behavior of the steady-state particle

density s , the survival probability P s , the correlation length

, and the characteristic time in the thermodynamic limit as

20

s , P s , − , − , 2

where is the distance from criticality.

At criticality the exponent characterizes the decay of

the density of particles t in the surviving samples when

starting with random initial conditions with finite density,

while for a single-particle initial condition the exponents ,

, and z characterize the survival probability P s t of the

processes, the number of particles Nt averaged over all

samples, and spreading distance Rt averaged over survival

samples, respectively, as 21

tt − , P s tt − , Rtt 1/z , Ntt . 3

Dynamical scaling implies the relations =/ ,

=/ , z= / . The well-known scaling relation between

these exponents in the usual APT is 21

= d/z − − .

4

In many systems and are equal 21 and hence =.

The critical behavior where criticality of all levels coincides

is characterized by a varying order parameter exponent

taking a different value in each level of the hierarchy.

Within the mean-field approximation, the exponent of a

level k is simply given by k =1/2 k with k0 for both the

coupled DP and PC processes 13,14. This k obtains corrections

below the upper critical dimension d C .

On the other hand, the exponents and remain unchanged

in all levels, but and vary according to the level.

However these exponents do not satisfy the ordinary scaling

relation Eq. 4. Instead for the coupled DP processes they

are suggested to satisfy the following relation in each level k

14:

k = d/z k − k − 1 ,

5

with k = k . The exponent 1 of P s t of the first level replaces

k of the level k. Generally k is the exponent for the

particle density at level k for random initial conditions at the

first level. In what follows we refer to the first level as source

and higher levels as slave levels.

B. Heterogeneous spreading and source average

The derivation of Eq. 5 from scaling arguments of 14

involves some tacit assumptions on taking averages which

may lead to wrong conclusions if ignored. It has been shown

that for the generalization of Eq. 5 to other couplings and

for understanding the appearance of 1 , it is necessary to take

the heterogeneity of active regions on the slave levels into

account and to apply a particular average method, the source

average 16. It turns out that the active regions of the slave

levels are divided into two regions, the so-called coupled and

uncoupled region. Careful analysis yields generalized scaling

relations consisting of two separate equations for the two

regions respectively 16. For self-containedness we briefly

review the significance of these notions.

1. Source average

By construction of the process, i.e., due to the feeding of

particles to higher levels, each level decays more slowly than

its lower levels. So the coupling to lower levels is always

broken after some time. From this time on the decoupled

level k evolves autonomously according to its own dynamics,

with critical exponents given by the pure uncoupled universality

class to which this level belongs. Therefore a trivial

change of universality class at level k occurs abruptly when

the lower level k−1 dies out. Thus no information about

different universality classes due to coupling can be gained

by studying realizations of the process with broken coupling.

In order to avoid this problem we only consider the configurations

in which the source k=1 still survives. In these

configurations, slave levels k1 always survive with unit

probability due to the coupling so the survival probability

P s t of the slave is just that of the source and we have the

following relation for any k1:

k k , k = 1 . 6

We call these configurations the source ensemble and the

average with this ensemble we refer to as the source average.

Of course, one may average dynamic quantities over the configurations

in which the level under consideration survives

regardless of the survival of other levels. This implies k

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C. Generalized scaling relations

FIG. 1. The evolution pattern of a cluster on the second level

starting from a single particle on the first level. The gray regions

outside the black regions are uncoupled.

1 for general couplings. We refer to this average as the

slave average, but we shall not employ it for the reasons

given above.

Our discussion not only explains the occurrence of 1 in

Eq. 5 for coupled DP systems but also suggests that 1 will

occur for any coupling, defined via the source average. The

source average more clearly exhibits the special properties of

unidirectionally coupled systems than the usual slave average

and it allows the derivation of the scaling relations without

any assumptions on scaling functions for observables.

2. Dynamic heterogeneity

The origin of the dynamic heterogeneity of the slave levels

can be understood as follows. Starting with a single particle

on the source level k=1, then in level k1 a cluster

of particles A k is formed by the coupling dynamics 1 and

it spreads according to the given evolution rules. Due to the

coupling, the average size of the cluster on each level is

always larger than those of the lower levels. It means that the

cluster of the level k with size R k t is divided into two parts,

the coupled and the uncoupled region. The coupled region of

the level k at time t is defined as the set of all sites within the

range between the leftmost and the rightmost particles of the

level k−1 at time t. The exterior of the coupled region is the

uncoupled region of the level k Fig. 1.

Inside the uncoupled region with size R U t=R k t

−R k−1 t, the dynamics on the level k evolve autonomously

without particle feeding from the level k−1. In the coupled

region with size R C t, the level k−1 feeds particles to the

level k so that R C t is actually the spreading distance R k−1 t

of the level k−1. Due to the coupling, the number of particles

N C t inside the coupled region increases more rapidly

than that of the uncoupled region N U t. This observation

suggests that the number of particles in the level k can increase

in time with different exponents, k C in the coupled

region and k U in the uncoupled region respectively, at the

multicritical point,

N C tt k C , N U tt k U . 7

We stress that even though inside the uncoupled region the

particle system evolves according to the critical dynamics of

level k, the critical exponents may be different. This is because

the boundaries of the uncoupled region and hence its

overall size and therefore its critical exponents are determined

by the coupling to the lower level. It is clear that this

heterogeneity of the particle distribution of the level kN k t

is a general property of unidirectionally coupled systems.

We discuss the hyperscaling in the level k in d dimensions,

using the source average. The index k is omitted from

the number of particle N and density in the coupled and

uncoupled region for simplicity of notation. In the coupled

region, N C t is given by the relation

N C t = C t R C d t P s k=1 t,

where C t is the particle density of the level k of surviving

samples inside of the volume R C d t and P s k=1 t is the survival

probability of the source level. Assuming algebraic decay

at the multicritical point C t − k C , R C =R k−1 t 1/z k−1, and

P s k=1 t − 1, respectively, we obtain the first scaling relation

k C = d/z k−1 − k C − 1 .

In the uncoupled region, assuming U tt − k U

t 1/z k U , we find the second relation

8

9

and R U

U k = d/z U k − U k − 1 .

10

The above two relations are generalized scaling relations

for unidirectionally coupled systems taking the heterogeneity

into account. They are very different from Eqs. 4 and 5

because they are not scaling relations for quantities of the

whole system. As N k t is given by N k t=N C k t+N U k t, the

scaling of N k t is determined by the scaling behavior of the

two distinct regions

k = max k C , k U .

11

For the scaling of the total size of the level kR k =R k U +R k−1

t 1/z k, we have the condition, z k U z k ,or

1/z k = max1/z k U ,1/z k−1 . 12

Similarly one has for the total density k tt − k the relation

k = min k C , k U .

13

Notice that for the special case of z k U =z k−1 , the exponent

k C is larger than k U because k C and k U satisfy the condition

k C k U . Therefore, we have two equalities of k = k C and

k = k C and we recover the scaling relation of Eq. 5 but

with k k . In general, as dynamic exponents such as k

and k depend on the average methods, the value of k of the

slave average differs from that of the source average.

D. Corrections to scaling

We remark that Eqs. 9 and 10 also quantify the corrections

to scaling of dynamic quantities coming from the generic

heterogeneity of coupled systems. The effective exponent

of N k t is defined as

k t =lnN k mt/N k t/ln m,

14

and similarly for k t and R k t with a positive m. The heterogeneity

induces corrections to scaling of the form t − ,

t − , and t − z with some negative exponents

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= C k − U k , = C k − U k , z = 1/z U k −1/z k−1 .

15

These correction exponents usually become very small and

therefore they cause a significant long time drift of the effective

exponents. These corrections explain the difficulties of

precise estimations of the dynamic exponents in unidirectionally

coupled systems.

III. DP-PC COUPLING

To simplify notation for two-level couplings we call particles

of the source k=1 and the slave level k=2 A and B,

respectively. Then for the slave level we have the scaling

relations from Eqs. 9 and 10

C =1/z A − C − A ,

U =1/z U − U − A ,

16

where we omit the index B from , , and z. Relations

11–13 become

= max C , U ,

17

1/z = max1/z U ,1/z A , 18

= min C , U .

19

For DP-PC coupling, we consider the contact process

CP and branching annihilating random walks with two offspring

BAW2 as typical particle models belonging to the

DP and PC class, respectively 22,23. In the CP of the

source level, a particle A is spontaneously annihilated with

rate p and it creates one A particle on one of the nearest

neighboring NN sites with rate 1−1−p. In branching

processes, if the target site is occupied by an other particle,

the branching is rejected. The rate is the coupling strength.

The criticality is p A c =0.232 6744 at =0 22. Using the

fact that the ratio of the creation and annihilation rates, R

=1−p c / p c , should be same at criticality for any value of ,

we find that the critical line of CPs in the − p phase diagram

is A c =1−pR A /1−p with R A =3.297 85.

In the BAW2 of the slave level, a particle B hops to one

of the NN sites with rate p and it creates two particles on two

NN sites to the left or right direction with 1−p. Iftwo

particles happen to be on a same site by branching or hopping,

they annihilate each other instantaneously. The critical

point of BAW2 at =1 is located at p B c =0.51057 23.

The critical line is B c = pR B /1−p, where R B =1−p B B

c / p c

=0.958 86.

The coupling dynamics of Eq. 1 now becomes

FIG. 2. Schematic phase diagram of two-level hierarchies with

coupling strength and interaction parameter p. The region labeled

by “Absorbing phase” corresponds to an absorbing phase of both

levels.

A → A + nB with 1−p, 20

where we take without loss of generality n=2.

The coupling dynamics 20 couples the CP and BAW2

without feedback from the BAW2 to the CP. An A particle

creates B particles on neighboring sites of the same site of

the BAW level only if the target sites are empty. The multicritical

point M , p M of the CP-BAW hierarchy is located at

M =0.225 26 and p M =0.190 23 within errors of p c A and p c B .

The well-known values of the exponents and 2/z of the

source level are A 0.159 and 2/z A 1.265 2.

Figure 2 shows the schematic general phase diagram of

two-level hierarchies. In region I, the two levels are both in

an active phase of nonzero steady-state particle density. In

region II, both levels are active but the slave level is slaved

to the first. The slave level undergoes the same type of phase

transition as the source level at the critical line of the source

level. In region III, two levels are completely decoupled.

While the source level is inactive, the slave level is active.

They undergo their own types of phase transition at their

respective critical points.

First we perform dynamic Monte Carlo simulations starting

with a single A particle on the source level at the multicritical

point M , p M . For the average of dynamic quantities

we take the source average. Using the effective exponents of

Eq. 14, we obtain asymptotic values of the exponents , ,

and 1/z of the slave level in the coupled and uncoupled

regions, respectively. We list all measured values of the exponents

for all couplings in Tables I and II.

Figure 3 shows effective exponents as functions of time of

the coupled uncoupled region and the whole active region.

The different values of the exponents in the both regions

clearly show the heterogeneity of the growing cluster generated

from a single A particle. By substituting the measured

values of C and U into Eq. 16, we predict C

=0.13310 and U =0.302 which agree very well with the

TABLE I. Measured values of critical exponents of the slave

level for the four couplings. The subscript C U corresponds to the

C U

coupled uncoupled region. pred and pred are the predicted values

from Eq. 16 using measured values.

Coupling C U C U C

pred

U

pred

DP-PC 0.131 0.281 0.341 0.141 0.341 0.162

DP-DP 0.081 0.161 0.401 0.311 0.391 0.312

PC-DP 0.142 0.161 0.142 0.1855 0.152 0.18520

PC-PC 0.211 0.2855 0.081 0.011 0.081 0.001

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TABLE II. Measured values of 2/z B and 2/z U for the four couplings.

The values of 2/z B,pred are estimated values from Eq. 18.

2/z DP-PC DP-DP PC-DP PC-PC

2/z U 1.202 1.272 1.262 1.151

2/z B 1.241 1.272 1.222 1.151

2/z B,pred 1.265 1.272 1.265 1.151

numerical estimates of Table I within errors. All dynamic

exponents satisfy the generalized relations of 16 well.

To obtain exponents of the total quantities B , N B and R B ,

we should compare the values of the coupled and uncoupled

region according to Eqs. 17–19. AsN C increases more

rapidly than N U C U , N B is expected to follow the scaling

of N C . It implies B = U . Similarly B and R B should

follow the scaling of C and R C =R A due to C U and

1/z C 1/z U respectively. So we expect B = C and z B =z A

for DP-PC coupling. However in Fig. 3, B t saturates to a

smaller value than C . The other exponents B t and 2/z B t

also show the same tendency. As mentioned in Sec. II, these

discrepancies come from the corrections to scaling with the

exponents defined in Eq. 15. Directly measuring the effective

slope of the correction to N B , viz., through the ratio

N B t/N C t, we obtain =0.201 which agrees with the

prediction = C − U =0.202. Similarly we also find correction

exponents =0.151 and z =0.031, which agree

FIG. 3. Effective exponents , , 2/z of the slave level for

DP-PC coupling. From top to bottom in each panel, each line corresponds

to the coupled, whole, and uncoupled regions,

respectively.

FIG. 4. Effective exponents of corrections to scalings of the

slave level for DP-PC coupling. From top to bottom, each line

corresponds to z , , and .

well with the predictions =0.151 and z =0.022, respectively.

Figure 4 shows the effective exponents of the

ratios. We should note that the corrections come from the

heterogeneity of a growing cluster. The effect of coupling

causes the long drift of effective exponents as shown in C t

especially.

Since the dynamic exponents and generally depend on

the average methods 6, the dynamic critical behavior

should be determined by and z. The exponents and z

defined in Eq. 3 are the ratios of static exponents , , and

in an active phase where all levels in the hierarchy survive

with finite probability. The finite survival probability of each

level implies the equivalence of source and slave average so

the static exponents are independent of average methods.

From this analysis we see that due to coupling with a DP

process the spreading properties of the PC process BAW2

completely change and show DP nature. It can be understood

in general terms from the rapid spreading of DP processes

with smaller z than that of the PC class z U z C . As the

typical sizes of DP vacuum domains and in an active

phase are smaller than those of the PC class, vacuum domains

of PC class shrink by the activation of the source level

and they should follow the scaling of the DP class. It implies

that and should be the values of the DP class so we

have z=z DP . So the DP-PC coupling completely changes the

scaling of both spatial and temporal correlation lengths, unlike

other couplings.

On the other hand, as the DP processes on the source level

survive longer than PC processes do, the coupling with the

source makes the decay of the total density B slow enough

to change the exponent in the coupled region C . The C

of Table I is even smaller than the value of the DP class,

DP 0.16. It is consistent with the fact that the exponent

changes and actually decreases as levels go up as in other

unidirectionally coupled systems. With the DP value of ,

we predict B = C DP =0.231.

For the measurement of the exponent of the slave level,

we measure B in the steady state B s with half-filled random

initial conditions on the source level. We average B s for

= p M − p=810 −2 –210 −4 and system sizes L=4096–1.2

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FIG. 5. Effective exponents eff of B s for DP-PC coupling.

10 4 Fig. 5. We find B =0.222 which agrees very well

with our prediction. We conclude that while the coupling

between the same type processes only changes the exponent

of higher levels, DP-PC coupling completely changes the

scaling behavior of the spatial and temporal correlation

length in addition to of the slave level.

IV. PC-PC COUPLING

In PC-PC coupling, we then consider BAW2 model in

both levels. Although the coupling was already studied 17,

we investigate the critical behavior in a more detailed and

transparent manner using the heterogeneity and source average.

In the source level, an A particle hops with rate p and

creates two offsprings on two nearest neighbor sites to the

left or to the right with rate 1−1−p. In the slave level,

hopping and branching of a B particle occur with rates p and

1−p, respectively. The two levels are unidirectionally

coupled with the coupling dynamics Eq. 20. The critical

lines of the two levels are c A =1−pR/1−p and c

B

= pR/1−p with R=0.958 86. The multicritical point is located

at M =1/2 and p M =0.342 732. The well-known exponents

and 2/z of the source level are A =0.2852 and

2/z A =1.1412 24.

Starting with two A particles on the source level, we measure

the effective exponents , , and 1/z in the coupled and

uncoupled regions Fig. 6. We performed simulations for

n=1 and 2. The two results are very similar so we present

results only for the more natural case n=2 Tables I and II.

As shown in Table I, the particle number of the coupled

region N C increases more rapidly so that the total particle

number N B follows the scaling of N C and we expect B

= C . Similarly we expect B = C for the total particle density.

Figure 6 shows a small deviation between the total and

the coupled regions, but it can be explained by the corrections

to scaling. From Eq. 15 we estimate correction exponents

=0.18510, and =0.071.

With the measured values of C ,1/z U given in the table

and A and z A of the PC class, we predict C =0.211 and

U =0.281 which agree well with the numerical results in

FIG. 6. Effective exponents of the slave level for PC-PC coupling.

From top to bottom in each panel, each line corresponds to

the coupled, whole, and uncoupled regions respectively.

Table I. Hence Eq. 16 is satisfied very well in PC-PC coupling.

The PC-PC coupling only changes C and C , like the

DP-DP coupling. As the spatial and temporal correlation

lengths follow the same scaling in both levels before the

coupling, the exponents and of both levels should be

the same after the coupling. This leads to z A =z B . Only the

density in the coupled region decays more slowly than in the

PC class. This implies the decrease of in the slave level.

We estimate B =0.681 with the relation B = C PC , which

is smaller than PC =0.923 24.

V. OTHER COUPLINGS

In this section we present some improved simulation results

on DP-DP and PC-DP couplings, respectively, and we

briefly illustrate the essential importance of using the source

average for the validity of hyperscaling relations for the

three-level PC-DP-DP couping. For DP-DP coupling, we

consider two CP models. In PC-DP coupling, A particles of

BAW2 evolve on the source level with hopping rate p and

two-offspring branching rate 1−1−p. In both couplings,

the two levels are coupled through the dynamics 20.

For the location of the corresponding critical points see 16.

In DP-DP coupling, effective exponents of the coupled

region saturate to asymptotic values as shown in Fig. 7,

which also well satisfy the scaling relation Eq. 16. The

same value of z in both coupled and uncoupled regions indicates

that the scaling of spatial and temporal correlation

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FIG. 7. Effective exponents of the slave level for DP-DP coupling.

From top to bottom in each panel, each line corresponds to

the coupled, whole, and uncoupled regions, respectively.

length remains unchanged as noted in Refs. 13,14. However,

the small value of C indicates the decrease of . With

the relation = C DP we estimate =0.141 which is compatible

with the result of 14, 2 =0.13215.

In PC-DP coupling, the DP level is expected to show DPtype

critical behavior because PC processes decay faster with

PC DP . The fast decay and slow spreading of PC processes

yields for the exponent of total particle number B

= U rather than C , unlike in other couplings. Assuming DP

critical behavior, we expect C =0.13, U =0.185, C = U

=0.16, and 2/z B =1.265. This is well borne out by the simulation

results, even though Fig. 8 shows that the exponents of

the coupled region converge to the expected values very

slowly. The long drift of C and C comes from the effect of

the coupling dynamics by itself, while the total spreading

distance R B t has strong corrections to scaling R C /R U

t − z.

With the values of and z in Tables I and II, Eq. 16

predicts C =0.152 and U =0.162, which agree with the

DP value DP =0.16 2. The scaling relation of Eq. 16 is

well satisfied for PC-DP coupling although the particle density

decays uniformly in the whole region of the slave level.

On the other hand, with the slave average, all exponents

including B should be DP values, which satisfy the ordinary

scaling relation of Eq. 4 rather than Eqs. 5 and 16. As

Eq. 5 is the result of Eq. 16, we conclude that the slave

average actually cannot satisfy the scaling relations of unidirectionally

coupled systems. However, the results of a slave

average satisfy Eq. 5 in other couplings. This contradiction

FIG. 8. Effective exponents of the slave level for PC-DP coupling.

From bottom to top in each panel, each line corresponds to

the coupled, whole, and uncoupled regions, respectively. t shows

differences only after 510 4 time steps.

can be explained by examining the composition of the slave

ensemble. In the slave average, as we only consider the configurations

in which the slave level under consideration survives,

the source level may survive or already have disappeared.

So the ensemble of the slave average includes some

configurations of source average which characterize the critical

behavior of higher levels. This is why results of the slave

average can satisfy Eq. 5.

The three-level hierarchy of PC-DP-DP coupling is another

example where the average method is crucial for the

validity of hyperscaling relations. In this hierarchy, the critical

behavior of the third level is affected by the second, but

the second is not affected by the first. So if we use the slave

average, the exponents of the third level should be those of

the slave level of DP-DP coupling. These exponents of the

slave level satisfy only the scaling relation of the two-level

hierarchy without the PC process in which 1 is the DP

value. However, as 1 in the scaling relation of the threelevel

hierarchy is the PC value, the exponents of slave average

cannot satisfy Eq. 5 of the PC-DP-DP coupling.

046122-7


S. KWON AND G. M. SCHÜTZ PHYSICAL REVIEW E 71, 046122 2005

VI. CONCLUSION

In summary, measuring static and dynamic exponents, we

investigate the change of critical behavior of two-level systems

in which the slave level is unidirectionally coupled to

the source level through the coupling dynamics A→A+nB.

Each level belongs to either the DP or PC universality class.

The coupling makes the active region of the slave level inhomogeneous.

Therefore the particle number exhibits different

scalings in two regions, the so-called coupled and uncoupled

regions. Proper analysis of the change of scaling

properties due to the coupling necessitates the use of the

source average. The heterogeneity of the particle distribution

in the slave level is characteristic for all unidirectionally

coupled systems and leads to generalized scaling relations

which are valid irrespective of whether the critical behavior

of higher levels is changed or not by given couplings.

However, it should be noted that the slave average may be

better than the source average in other regions of the phase

diagram Fig. 2. This comes from the fact that except at the

multicritical point, the slave levels are completely uniform

and we cannot divide the whole active region into two distinct

regions. For example, in region III of the generic phase

diagram, the two levels are completely decoupled and the

source cannot affect the critical behavior of the slave because

the source is inactive. So we cannot define the coupled region

which is the characteristic of unidirectionally coupled

systems at their common critical point. In region II, the slave

level is completely slaved to the source so the whole active

region of the slave level is the coupled region. In this region

we cannot define the uncoupled region and we have just one

scaling relation of the coupled region.

At the multicritical point the critical behavior strongly

depends on that of the source level. For DP-DP and PC-PC

couplings, the coupling only changes the scaling of the order

parameter. The scalings of spatial and temporal correlation

lengths do not change. For DP-PC coupling, the DP level

completely changes the critical behavior of the PC level.

While the spatial and temporal correlation lengths follow the

scaling of the DP class, the order parameter exhibits slower

decay. However, in PC-DP coupling, the PC level is too

weak to change the critical behavior of the DP level.

These results for the four couplings suggest the following

simple criterion for the critical behavior of the slave level in

unidirectionally coupled two-level hierarchies.

1 The coupling of models belonging to the same class

only changes the scaling of the order parameter as already

noted in 13,14,17.

2 The coupling of more slowly with more quickly

spreading systems such as DP-PC coupling completely

changes the critical behavior of the slave level. While the

spatial and temporal correlation lengths follow the scaling of

the source level, the order parameter decays more slowly

with a smaller exponent B than that of the source.

3 Finally, the coupling of faster decaying with more

slowly spreading systems such as PC-DP coupling does not

change the critical behavior of the slave. The coupling only

makes the slave converge to its asymptotic critical behavior

very slowly and it causes a long drift of effective exponents.

Since in unidirectionally coupled systems of more than

two levels each level is affected by the level below, it is

possible to apply the above criterion of two-level hierarchies

to general hierarchies.

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